introduction to phase transitions

15.06.2009 Phase Transitions UHD 1

Introduction to

Phase Transitions

D. Dubbers

• Particle physics Standard model …• Nuclear physics Quark-gluon transition

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Phase transitions in Heidelberg physics dep't.:

1. Introduction

History of the universe

of the vacuum:

Transition Temperature TimePlanck 1019eV ~0sGUT’s ? ?Inflation ? ?Electro-weak 100GeV 10−12s

of matter, i.e. freeze out of:

Quark-gluon plasma to nucleons 100GeV? 10−12s ? Nucleons to nuclei 1MeV 1sAtoms 10eV 105aGalaxies 3K today

= Succession of phase transitions

Topics not treatedother phase transitions:

Bose-Einstein condensatesSuperfluidity

Quantum phase transitionsAggregatesFragmentation PercolationLiquid crystalsIsolator-metal transitionsTopological defectsTraffic jams…

other 'critical phenomena' (non-linear physics)Route to chaosTurbulenceSelf organized criticality (forest fires, avalanches, …)…

2. Phenomenology: control parameter

Phase transition = sudden change of the state of a system upon variation of an external control-parameter:which reaches 'critical value'.

Here: control-parameter is temperature(it can also be pressure, atomic composition, connectivity, traffic density, public mood, taxation rate, …):

example: magnetTC = critical temperature:

above TC: paramagnet PMbelow TC: ferromagnet FM

here: TC = Curie temperature

critical phenomena

Why are phase transistions so sudden?

example: liquid

below TC: liquid Labove TC: gas G

Example of boiling water:bond between 2 molecules breaks due to thermal fluctuationincreased probability that 2nd bond breaks, too: chain reaction

below TC: broken bond heals, before 2nd bond breaks – water in boiler is noisyabove TC: broken bond does not heal, before 2nd bond breaks – water boils:

A 'run-away' or 'critical' phenomenon: L → G

order parameter

At TC the probe acquires a new property, the order-parameter M:

above TC: M=0. below TC: M≠0,

here = magnetisation M

N.B.: Disorder: high symmetry: S ↓ ↓ UOrder: low symmetry: S'

PM: rotational symmetry UFM: cylindrical symmetry

critical exponent

Observation:

At T <≈ TC the order M parameter depends on temperature T like:

below TC: M(T ) = M0 (1−T/TC)β

with critical exponent β

Examples: M(T ) ~√TC−T :

critical exponent β = ½

M(T )~3rd√TC−T :

critical exponent β = ⅓

1-dimensional magnet:"bifurcation"

FM PMT

reduced temperature

With reduced temperature

t = (TC−T)/TC

and m = M/M0:

the order parameter scales with temperature

like m = tβ,

or ln m = β ln t

Temperature dependence of magnetisation measured by magnetic scattering of neutrons

log-log linear-linear

six main critical exponents

critical expon't:1. Order parameter m = tβ β

2. Susceptibility χ = | t |−γ γ

3. Critical order param. Mc = | h |1/δ δ

4. Specific heat C = | t |−α α

5. Coherence length ξ = | t |−ν ν 6. Correlation function G = 1/r d−2+η η

χ FM PM χ− χ+

critical magnetization →

ξ, τ FM PM

P Exp.:

water:

universality

All systems belonging to the same universality class have the same critical exponents.

Example: Water near critical point and (1-dim) Magnet

What defines a universality class?

latent heat

Heat a block of ice:

Melting S → L

Transition: order → short range order

Boiling L → G

Transition: short range order → disorder

Breaking of bonds requires energy =

latent heat = difference in electrostatic potential,

without change in kinetic energy (temperature).

At critical temperature TC:

Addition of heat only changes ratios ice/water or water/vapor, but not the temperature

--S-| |---L---| |-G--

boiling: Tb=1000→

melting Tm= 00→

| Qb |

Water (H2O):

///e e e potkinE kTE kTE kTP

divergence of heat capacity

T / 0C

C=dQ/dT

1st and 2nd order phase transitions

Latent heat: Qb = ∫L→GPdV = area in P-V diagr.

When latent heat: Qb > 0: 1st-order phase transition.

At the critical point latent heat Qb = 0: Continuous phase transition(or 2nd order phase

transition)

Boiling water:Order parameter = ρliquid−ρgas

p-V phase diagram for water (H2O):

| order param. |

↑ Vc

equation of state

Equations of State: describes reaction to variation of external parameters:

Pressure P = P(V,T,…)Magnetization M = M(B,T,…)

Example:

Ideal gas: P = RT/V = ρkT Gas equation

(ρ = NA/V, R = NAk)

Real gas: (P +a/V2)(V−b) = RT attractive ↑ ↑ repulsive potential

van der Waals-equation

same in P-ρ diagram:

G L+G L

universality of v.d.W.-equation Bild Yeomans p. 28:

Reduced van der Waals-equation:

(P/PC +3(VC/V)2) (V/VC−1/3) = 8RTC

with critical values PC, VC, TC

seems to be universal

even far away from TC.

G G+L L

ρ/ρC

free energy → everything else

Yeomans p.17: (β=1/kT)master plan

example: paramagnetism

spin ½:

energy/molecule E± = ±μB

partition function for N molecules:

β=1/kT

magnetisation

Saturation magnetis. M0 = Nμ

Susceptibility χ = ∂M/∂B ≈ Nμ2/kT: χ ~ 1/T

= Curie Law, for kT >> μB

E eeeZ r )(

3. Landau model

Landau 1930 (Landau-Lifschitz 5: Statistical Physics ch. XIV) 1-dim magnet:

'Landau' free energy of ferromagnet F = F(m)

with magnetization m = <M>/M0 (mean field approx.)

Taylor-expanded about m = 0:

F = F0(T) + (½a' m2 + ¼λ m4)V

(only even powers of m, λ > 0)

2-dim magnet:

a' changes sign at T = TC: a' = a·(T−TC):

Free energy density f = (F − F0)/V then is:

f = ½a(T−TC) m2 + ¼λ m4

Spontaneous magnetization in zero external field

Landau: f = ½a(T−TC) m2 + ¼λm4

At equilibrium → minimum of free energy:

∂f/∂m = a(T−TC) m + λm3 = 0

and a(T−TC) + λm2 = 0

for T≥TC: Magnetization m = 0: PM

for T<TC: Magnetization m = ±(a/λ)½(TC−T)½ FM

first critical exponent β = ½

Same result for order parameter of v.d.W. gas:

ρL − ρG ~ (TC − T)½.

L+G Δρ 2 phases 1 phase

Magnetization in external field

Magnetic energy in external magnetic field h:

f = ½a(T−TC) m2 + ¼λm4 − hm

At equilibrium: from

∂f/∂m = a(T−TC) m + λm3 − h = 0

follows magnetization ±m(T), see Fig.,

in particular:

critical magnetization at T = TC:

h = λm3

third critical exponent δ = 3

Same result for critical isotherm parameter of v.d.W. gas:

m FM PM

Th=0↑ ↓h>0

↑h<0

critical magnetization →

Magnetic susceptibility

Susceptibility χ = ∂m/∂h diverges at T=TC

Reason: Free energy has flat bottom at T=TC:

above TC: χ+ = [a(T−TC)]−1 PM

below TC: χ− = [2a(TC−T)]−1 = ½χ+ FM

Curie-Weiss law

second critical exponent γ = 1

Same result for v.d.W.-compressibility κ− and κ+:

χ FM PM χ− χ+

κ L G κ− κ+

Specific heat in zero field

Entropy density

s−s0(T) = −∂f/∂T

Specific heat

c−c0(T) = −∂s/∂T

Result: Specific heat makes a jump at TC:

fourth critical exponent α = 0

Same result as for specific heat of v.d.W. gas:

c−c0(T)

s−s0(T)

Compare with experimentMagnetization: M ~ (TC−T)β

Landau:

c M χ Mc

specific heat has a small α>0

4. Ginzburg-Landau theory of superconductivity

Non-uniform superconductor (mixed phases, Meissner effect, etc.):

order parameter = cooper pair wavefct. is position dependent: ψ=ψ(r)

like in elasticity theory:

energy penalty for deviations from homogeneity is ~ |ψ|2.

Free energy:

Fs = Fn + ∫V ((ħ2/2m*) |ψ|2 + ½μ2·(T −Tc) |ψ|2 + ¼λ|ψ|4)dV

Ekin + Epot

(Ginzburg-Landau, 1950)

superconductor in magnetic field

B = B(r) = ×A(r), with vector potential A,

A changes momentum mυ of a particle to

p = mυ + eA

but does not change its energy

E = (mυ2)/2m = (p−eA)2/2m

so for B ≠ 0

Fs = Fn + ∫V (|−iħψs−e*Aψ|2 /2m* + ½μ2·(T −Tc) |ψs|2 + ¼λ|ψs|4 + B2/2μ0 − B·M) dV

Tsc + Vsc + Efield + Emagn

m*=2me, e*=2e

Lit.: C.P. Poole et al.: Superconductivity, ch.5, Academic Press 1995

two scales in superconductivity

Mean field theory of superconductivity (Ginzburg-Landau):

Superconductor has 2 characteristic scales:

1. of the order parameter = superconducting condensate ψ:

coherence length ξ = ħ/μc of the condensate

2. of the magnetic field, via the Meissner effect:

London penetration depth λL = ħ/(eυc)

vacuum supercond.

x ξ=ħ/μc

Bvacuum supercond B0

λL =ħ/mphc

Superconductivity and Standard Model

: tensorsfield

1D derivativecovariant with

1)(¼)(½)D()D(

density Lagrange Salam-Weinberg:mechanism Higgs ,

,i ,2 ,2 :condensateCooper * derivativecovariant with

density LagrangeLandau -Gintzburg:effect Ochsenfeld-Meissner ,

:ichenWir vergle

ig'BτigA

WWΦΦλΦΦμΦΦL

ψψψee*mm*ie

Bψλψμψ

μμμμ

μνμνμμ

model Standard

ctorSupercondu

B = ×A

Comparison of coefficients:

G.-L.: Uel-mag(1) W.-S.: SUL(2)×UY(1)

order parameter: super-conducting condensateψ = ψ1 + iψ2

Higgs doublet Φ1 + iΦ2

Φ = Φ3 + iΦ4

boson mass generationby Higgs field:

Meissner effectmph = e <ψ1>

Higgs mechanismmW = g <Φ3>

Compton wavelength λ of interacting boson:

London penetration depthλL= ħ/(mphc)

range of weak interaction λW=ħ/(mWc)

Compton wavelength λ of Higgs:

coherence lengthξ = ħ/(μc)

"coherence length"λH= ħ/(mHc)

N.B.: die maximale Reichweite einer WW durch virtuellen Austausch ist R = cΔt = ħc/E0 = ħ/mc ≡ λCompton

Dasselbe im Detail:

Im Folgenden zeigen wir:

1. das Entstehen von Goldstone Bosonen anhand der Magnonen-Anregung in einem Ferromagneten im Landau Modell (spontane Brechung einer globalen Symmetrie)

2. den Higgs-Mechanismus, dh. das Verschwinden des Goldstone Bosons und die Entstehung der Masse des Eichbosonsanhand eines Supraleiters im Landau-Gintzburg Modell(spontane Brechung einer lokalen Eichsymmetrie)

Goldstone's theorem for Landau magnetGoldstone's theorem:

Each spontaneous breaking of a continuous symmetry

creates a massless particle (i.e. an excitation without an energy gap)

= Goldstone Boson

Simple example: Landau ferromagnet: like in elasticity theory:

energy penalty for deviations from homogeneity is ~ |m|2.

solution above TC is rotationally symmetric:

solution below TC is cylindrically symmetric:

Goldstone mode belonging to broken symmetry = magnon

magnon dispersion relation:

magnon:

2 42 21 12 4( )Cf m T T m m

Magnon = Goldstone of broken PM→FM symm.

Ordnunghöherer WW.

Massenterm ohne Goldstone ½

längs ndGrundzustaneuen um Anregung massive"" ½ ½

Terme eunabhängig von const.:ergibt in eingesetzt

Minimum) imdet verschwin½ Term lineare in (der

½ const. ½ ½ :und

) i (wegen )( ½ ½

i ½ :n Fluktuatio

e )( /)(ie ½ ½ :lKettenrege

e ))(()( : Phaseder n Fluktuatio und )( Amplitudeder n Fluktuatio

: Minimum neues umnen Fluktuatio :

½ ½ ½

222222

2/)(i/)(i2

keEnergielüc ohne Magnon

MυMχχμχ

υχμχ

χμχυμMμ

babaxχ

χυχ

χυυxχM

xχυxMMxχ

MλMμML

υxυx

↑<M> = υ

spontaneous symm.breaking in superconductorGauge invariance requires interaction with massless field Aμ.

However, in a superconductor, the photon Aμ becomes massive.

Still: Ginzburg-Landau model is gauge invariant (Dr.-thesis Ginzburg ~ 1950)

The reason is what is now called the Higgs mechanism:

When a scalar, gauge invariant field ψ

suffers a spontaneous symmetry breaking,

then the vectorfield Aμ can become massive,

without losing its gauge invariance,

while at the same time the Goldstone disappears.

Ginzburg-Landau superconductor: Cooper pairs = Higgs field ψ:

Ls = Ln + |ψ−i2eAψ|2 −½μ2|ψ|2−¼λ|ψ|4 − B2/μ0* + B·M

with charge of Cooper pairs e*=2e.

Higgs-mechanism for superconductor

As before: Fluctuations of ψ(x) about <ψ> = υ at "bottom of bottle"ψ(x) = (υ + χ(x)) exp(iθ(x)/υ)

Then |ψ−ie*Aψ|2 = |χ + i(υ+χ)(θ/υ−e*A)|2, with χ<<υ:

if we choose gauge to A=A'+θ/e*υ, this becomes ≈ (χ)2 −υ2e*2A2,

and the massless Goldstone term (θ)2 disappears,

and the photon A becomes massive (but remains gauge invariant):

Ls = const. + (ħ2/2m*) (χ)2 −½μ2|χ|2 = "Higgs" with mass μ

−mph2A2 = heavy photon with mass mph=υe* = (a·(T−TC)/λ)½ 2e

− B2/2μ0 + B·M = field terms as before

+ some residual terms

The coherence length found before turns out to be ξ = 1/μ (ħ=c=1),

or ξ = ħ/μc = Compton wave length of the Higgs of mass μ =(4ma·(TC−T))½,

and the London penetration depth λL = 1/mph, or λL = ħ/mphc

= Compton wave length of the heavy photon

N.B.: number of degrees of freedom remains the same

υ χ ψ1

5. Fluctuations

Ginzburg-Landau and Weinberg-Salam are mean-field theories,which is sufficient.

But: in general time-dependent fluctuations must be included.

Correlation length ξ ~ mean size of a region of same densityCorrelation time τ ~ mean time of existence of such a region

When T → TC, then density fluctuations on all length scales and all time scales

Divergence: ξ → ∞, τ → ∞

MOVIE: CRITICAL SCATTERING

ξ, τ L G

P Exp.:

Critical fluctuations

Fluctuations are intimately linked to the susceptibilities

In a magnet: mean square fluctuations of magnetization:<M2> − <M>2 = kT·χ, with magnetic susceptibility χ.

At the Curie temperature: T=TC the critical magnetic fluctuations diverge like the static susceptibility χ,therefore critical exponents cannot be all independent of each other.

(same in liquid: density fluctuations are linked to compressibility κ)

(cf: dissipation-fluctuation theorem, Nyquist theorem)

Measurement of magnetic critical opalescence

Staatsexamens-Arbeit N. Thake 1999

χ+(T) ~ (T−TC)−γ

χ−(T) = ½ χ+(T)

Correlation function near the critical point

For many systems the spatial correlation function decays with distance r like:

G(r) ~ exp(−r/ξ)/rn

Near the critical point the correlation length ξ diverges like

ξ ~ |T−TC|−ν,

and the correlation function becomes G(r) ~ 1/rd+2−η,

with dimension d, and with two further critical exponents ν and η.

only 2 independent critical exponents

Empirically:

Only 2 of the 6 critical exponents are independent,

since the following 4 empirical relations hold:

α + 2β +γ = 2

α + β(1+δ) = 2

γ = (2−η)ν

dν = 2−α

What is the deeper reason for this?

Scale invariance and renormalizationMean field: Averaging over the fluctuations is not permitted

because fluctuation amplitudes diverge at the critical point.

Way out: successive averaging, separately for each scale,

starting with a small length scale L << coherence length ξ

(when working in real space).

Example for d=2 dimensional ('block-spin') iteration process:

Divide systems in blocks of volume Ld = 32 = 9 cells.

1. Take a majority vote in each block.

2. Combine the cells in a block and assign the majority vote to the cell.

3. Shrink new cells to the size of the original cells and renumber them. Number of configurations shrinks from 29=512 to 21=2.

4. 'Renormalize' the interaction Ĥ between the averaged elements such that the new partition function stays the same:

ZN'' = ∑2^N' config. e−βĤ' = ∑2^N config. e−βĤ = ZN ,

so that the physics remains the same (scale invariance). Go to 1.

+ + −

− + +

+ − −

+ + −

− + +

+ − −

+ − + + − − + + −

− − + − + + − + −

+ + − − + + − −

− + − + − − − + −

+ − − + + + + − +

+ + − − − + + − +

+ − + − + − − + −

− + − + + − + + −

+ − + + − − + − −

Block-spin operation in 2-dim.

Block-spin operation in 2-dim.: T=TC

Transformation of reduced temperature and field

At each iteration step:

coherence length shrinks from ξ to ξ ' = ξ /L,

that is temperature T moves away fromTC,

either to higher T → ∞ or to lower T → 0 temperatures:

Under an iteration the reduced temperature t = |(T−TC)/TC| changes

from t to t'= g(L) t, the function g(L) is to be determined:

Upon two iterations, successive shrinking is by L1, then by L2, in total by L1L2.

Reduced temperature changes to t' = g(L2) g(L1) t = g(L1L2) t.

A function with the property g(L2) g(L1) = g(L1L2) necessarily has the form g(L) = L y,

Check: L1y L2

y = (L1L2) y.

Hence the reduced temperature t transforms as: t' = L y t with exponent y>0.

Same argument for magnetic field: it increases when coherence length shrinks:

i.e. reduced field h transforms as: h' = Lx h with exponent x>0.

Critical exponent relations from scaling

Order parameter magnetization:

m = −∂f (t, h), /∂h|h→0

= L−d Lx ∂f (Ly t, Lx h)/∂h|h→0;

this holds for any L, in particular for

|Ly t| = 1, i.e. L = t−1/y:

m = | t |(d−x)/y ∂f (±1, 0)/∂h = const. | t |β,

with critical exponent β = (d−x)/y.

Critical exponent relations from scaling

With similar arguments:

2. Susceptibility χ = −∂2f (t, h)/∂h2| h→0 ~ | t |−γ,

with critical exponent γ = (2x−d)/y

3. Critical isotherm m = − ∂f (t, h)/∂h| t→0 ~ | h |1/δ

with critical exponent δ = x/(d−x)

4. Specific heat (h=0) CV = − ∂2f(t, 0)/∂t2 ~ | t |−α

with critical exponent α = 2−d/y

5. Coherence length ξ ~ | t |−ν

with critical exponent ν =1/y

6. Correlation function G ~ 1/r d−2+η

with critical exponent η = 2+d−2x

which can in principle be resolved to write all

critical exponents as functions of two variables x and y.

universality classes

The critical exponents depend on only two paramters x and y.

Can these take any value?

No, because they can be shown to depend only on

two other geometrical entities:

1. the spatial dimensionality d of the system

2. the dimensionality n of the order parameter

Example: Magnetization M:

n = 1: Ising model sz = ±1 in d = 1, 2, 3 dimensions

n = 2: xy-model with planar spin Mxy moving in x-y plane

n = 3: Heisenberg model with 3-vector M.

As d and n are discrete numbers, there is a countable number of

universality classes (d, n), and within each class the

critical behaviour in continuous phase transitions is identical.

Values of the critical exponents for (d, n)

with ε = 4 − d:

The higher the dimension, the less the system is disturbed by fluctuations.(example: Domino in various dimensions)For d = 4, we are back at the mean field results.

Critical exponents β and γ

various universality classes

Literature

J.M. Yeomans: Statistical mechanics of phase transitionsOxford 1992, 144 S, ca. 60 €readable and compact

P.M. Chaikin, T.C. Lubensky: Principles of condensed matter physicsCambridge 1995, 684 S, ca. 50 €concise, almost exclusively on phase transitions

I.D. Lawrie: A unified grand tour of theoretical physicsBristol 1990, 371 S, ca. 50 €really grand tour with many analogies

P. Davies: The New PhysicsCambridge 1989, 500 S, ca. 50 €in-bed reading

other sources will be given 'on the ride'

introduction to phase transitions

critical temperature

liquidbelow tc

noisyabove tc

critical order param

critical phenomenon

critical point

critical value

liquid labove tc

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